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CHAPTER 6
TERM STRUCTURE OF INTEREST RATES
6.1 INTRODUCTION
Maturity is one of the important factor in determining a bond's yield. In the financial
literature, the relationship between the rates of return on financial assets and their maturities
is referred to as the term structure of interest rates. Term structure is often depicted
graphically by a yield curve. Yield curves are plots of rates of return against maturities for
bonds which are otherwise alike.
These curves can be constructed from current
observations. For example, one could take all outstanding corporate bonds from a group in
which the bonds are almost identical in all respects except their maturities, then generate the
current yield curve by plotting each bond's YTM against its maturity. If investors are more
interested in long-run average yields instead of current ones, the yield curve can be
generated by taking the average yields over a sample period (e.g., 5-year averages) and
plotting these averages against their maturities. The major difficulty in generating such
curves -- regardless of whether they are current or average yields -- is obtaining a large
enough sample of bonds which are almost identical in terms of risk, liquidity, coupon yields,
and the like. To address this problem, a widely-used approach is to generate a spot yield
curve from spot rates (discussed in the last chapter) using Treasury securities.
Whether they are derived from current, spot, or averages rates, empirically generated
yield curves have tended to take on one of the three shapes shown in Exhibit 6.1-1. Yield
curves can be positively-sloped with long-term rates being greater than shorter-term ones.
Such yield curves are called normal or upward sloping curves. They are usually convex
from below, with the YTMs flattening out at higher maturities. Yield curves can also be
negatively-sloped, with short-term rates greater than long-term ones. These curves are
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known as inverted or downward sloping yield curves. Like normal curves, these curves
also tend to be convex, with the yields flattening out at the higher maturities. Finally, yield
curves can be relatively flat, with YTMs being invariant to maturity, or they can have both
positively-sloped and negatively-sloped portions, known as a humped yield curve.
The actual shape of the yield curve depends on the types of bonds under consideration
(e.g., AAA bond versus B bond), on economic conditions (e.g., economic growth or
recession, tight monetary conditions, etc.), on the maturity preferences of investors and
borrowers, and on investors' and borrowers' expectations about future rates. Four theories
have evolved over the years to try to explain the shapes of yield curves: Market
Segmentation Theory (MST), Preferred Habitat Theory (PHT), Unbiased Expectation
Theory (UET), and Liquidity Preference Theory (LPT). As we will see, each of these
theories by itself is usually not sufficient to explain the shape of a yield curve; rather, the full
explanation underlying the structure of interest rates depends on elements of all four
theories.
6.2 MARKET SEGMENTATION THEORY
Market Segmentation Theory (MST) posits that investors and borrowers have strong
maturity preferences which they try to attain when they invest in or issue fixed income
securities. As a result of these preferences, the financial markets, according to MST, are
segmented into a number of smaller markets, with supply and demand forces unique to each
segment determining the equilibrium yields for each segment. Thus according to MST, the
major factors which determine the interest rate for a maturity segment are supply and
demand conditions unique to the maturity segment. For example, the yield curve for high
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quality corporate bonds could be segmented into three markets: short-term, intermediateterm, and long-term. The supply of short-term corporate bonds, such as CP, would depend
on business demand for short-term assets such as inventories, accounts receivables, and the
like, while the demand for short-term corporate bonds would emanate from investors
looking to invest their excess cash for short periods. The demand for short-term bonds by
investors and the supply of such bonds by corporations would ultimately determine the rate
on short-term corporate bonds. Similarly, the supplies of intermediate and long-term bonds
would come from corporations trying to finance their intermediate and long-term assets
(plant expansion, equipment purchases, acquisitions, etc.), while the demand for such bonds
would come from investors, either directly or indirectly through institutions (e.g., pension
funds, mutual funds, insurance companies, etc.), who have long-term liabilities. The supply
and demand for intermediate funds would, in turn, determine the equilibrium rates on such
bonds, while the supply and demand for long-term bonds would determine the equilibrium
rates on long-term debt securities.
Important to MST is the idea of unique or independent markets. According to MST,
the short-term bond market is unaffected by rates determined in the intermediate or
long-term markets, and vice versa. This independence assumption is based on the premise
that investors and borrowers have a strong need to match the maturities of their assets and
liabilities. For example, an oil company building a refinery with an estimated life of 20
years would prefer to finance that asset by selling a 20-year bond. If the company were to
finance with a 10-year note, for example, it would be exposed to market risk in which it
would have to raise new funds at an uncertain rate at the end of ten years. Similarly, a life
insurance company with an anticipated liability in 15 years would prefer to invest its
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premiums in 15-year bonds; a money market manager with excess funds for 90 days would
prefer to hedge by investing in a money market security; a corporation financing its accounts
receivable would prefer to finance the receivables by selling short-term securities.
Moreover, according to MST, the desire by investors and borrowers to avoid market risk
leads to hedging practices which tend to segment the markets for bonds of different
maturities.
6.2.1 MST in Terms of Supply and Demand Curves
One of the best ways to understand how market forces determine the shape of yield curves is
to examine MST using fundamental economic supply and demand analysis. To see this,
consider a simple economic world in which two types of corporate bonds -- long-term (BLT)
and short-term (BST) -- and two types of government bonds -- long-term (BGLT) and shortterm (BGST) -- exist. At a given point in time, the supplies of these bonds being held by
investors in the market would be fixed. The fixed supplies of outstanding corporate bonds is
depicted graphically in Exhibit 6.2-1 by the vertical short-term corporate bond curve which
shows investors holding BSST short term corporate bonds and by the vertical long-term
corporate bond curve showing investors holding BSLT long-term bonds.
The vertical shape of the curves simply indicates that the amount of bonds outstanding
are fixed regardless of the rates on such bonds. While the supplies of bonds being held by
investors are fixed at given point in time, they do change as bonds in one maturity segment
in one period move to a lower maturity segment in the next period (e.g., 10-year bonds
becoming nine-year bonds as we move from one year to the next), as new bonds are issued
to finance new capital formations or refinance debt, and as bonds are called.
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A number of economic scenarios can be advanced to explain the changes in outstanding
bonds from one period to the next. One factor important in determining changes in the
amount of bonds outstanding is the overall state of the economy. In general, when an
economy is growing, businesses tend to expand both their short-term and long-term assets;
that is, in expansionary periods, companies often increase inventories in anticipation of
greater sales, experience increases in their accounts receivables, and often increase their
long-term investments in equipment, plants, new products, and marketing areas. As a result,
to finance the increases in short-term and long-term assets, corporations tend to issue more
securities in periods of economic growth. Moreover, the increase in new securities during
such periods tends to cause the total amount of bonds outstanding to be greater in that period
than in slower growth or recessionary periods. In contrast, during a recession, corporations
tend to maintain smaller inventories, realize fewer accounts receivables, and decrease their
long-term real investments. Consequently, the outstanding supplies of both short-term and
long-term corporate bonds are usually less in recessionary periods than in periods of
economic growth. It seems reasonable to assume that in periods of economic expansion the
supplies of both short-term and long-term corporate bonds held by investors are greater than
in periods of economic recession. Thus, in our model, both the short-term and long-term
vertical bond supply curves in Exhibit 6.2-1 shift to the right when the economy moves from
recession to expansion, and to the left when the economy moves from expansion to
recession.
While the amount of bonds outstanding depends on the state of the economy, investor
demand for bonds depends on their yield relative to the yields on substitute securities. For
this model, let us assume that the demand for short-term corporate bonds is greater when
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short-term yields (rST) are greater (or when short-term prices are lower) and when the yields
on short-term government securities (rGST) are lower. Similarly, for long-term corporate
bonds let us assume investors have a greater demand for such bonds when their yields (rLT)
are higher and the yields on long-term government bonds (rGLT) are lower. These bond
demand relations are depicted graphically in Exhibits 6.2-1 by the positively-sloped bond
demand curves BD BD. Each curve shows the direct relationship between the demand for
the bond and its YTM, with the YTM on the substitute government security assumed fixed.
For the short-term corporate bonds, if the YTM on short-term government bonds were to
increase, investors would want more government bonds and fewer short-term corporate
bonds. This would cause the short-term corporate bond demand curve to shift to left,
reflecting a lower demand for short-term corporate bonds at each rate. Conversely, if shortterm government rates were to decrease, the short-term corporate bond demand curve would
shift to the right, reflecting the greater demand for short-term corporate bonds at each yield,
given the lower yields on short-term government securities. The same analysis would apply
to the long-term corporate bond demand curve if long-term government rates were to
change. Note, in this economic model we are not assuming that the demand for short-term
bonds depends on long-term rates or that demand for long-term bonds is a function of shortterm rates. The absence of these variables is consistent with the MST assumption that
markets with different maturity segments are independent.
Given the factors determining the supply and demand for corporate bonds, the rate that
ultimately prevails in the market should be the one at which there is no excess supply or
demand. This equilibrium rate, r*, is graphically defined by the intersection of the supply
and demand curves. If the yield in the short-term corporate market, for example, were
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above this equilibrium rate, then investors would want more short-term corporate bonds than
they currently are holding. This excess demand would drive the price of the bonds up,
decreasing the demand (movement down along the demand curve) until the excess was
eliminated. On the other hand, if the YTM on short-term corporate bonds were lower than
its equilibrium, then bondholders would want to sell some of their bonds given their lower
rates and higher prices. This excess supply in the market would tend to lower prices and
increase yields (movement up the bond demand curve) until the excess supply was
eliminated.
Thus, only at r*, where bond demand equals bond supply, is there an
equilibrium where bondholders do not want to change.
6.2.2 Yield Curves Based on Supply and Demand Model
The equilibrium rates for short-term corporate bonds and long-term bonds, shown in
Exhibits 6.2-1, can be plotted against their corresponding maturities (simply denoted as S-T
and L-T) to generate a yield curve. The resulting yield curve is illustrated in the lower graph
in Exhibit 6.2-1. In general, the position and the shape of the curve depends on the factors
that determine supply and demand for short-term and long-term bonds. In our model, the
state of the economy determines the positions of the supply curves, and rates on government
securities determine the positions of the bond demand curves. These two factors help
explain why yield curves change. Several cases of yield curve changes are discussed below.
Case 1: Economic Recession
Suppose the economy moved from a period of economic growth into a recession. As noted,
when an economy moves into a recession, business demand for short-term and long-term
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assets tends to decrease. As a result, many companies find themselves selling fewer shortterm bonds (e.g., CP), given that they plan to maintain smaller inventories and expect to
have fewer accounts receivables. They also find themselves selling fewer long-term bonds,
given that they tend to cut planned investments in plants, equipment, and other long-term
assets. In the bond market, these actions cause the short-term and the long-term supplies of
bonds outstanding to decrease as the economy moves from growth to recession. At the
initial rate, the decrease in bonds outstanding creates an excess demand, with bondholders
now competing to buy fewer available bonds. This drives bond prices up and the YTMs
down, decreasing demand until a new equilibrium rate is attained.
This excess demand is reflected in leftward shifts of the short-term and long-term bond
supply curves, with lower bond supplies creating excess demand for short-term and longterm bonds (see Exhibit 6.2-2). The excess demands at rST* and rLT* are eliminated once
rates fall to their new equilibrium levels at rST* and rLT*. Thus, in the context of our model,
a recession has a tendency to decrease both short-term and long-term rates, causing the yield
curve to shift down as shown in Exhibit 6.2-2. Note that with this model we cannot explain
whether or not the slope of the yield curve also changes. To address that question we need
information about the magnitudes of the shifts of the long-term and short-term supply
curves, as well as the relative slopes of the demand curves. Finally, note that the opposite
results would occur if the economy moved from recession to economic expansion. This
analysis is left to the reader.
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Case 2: Treasury Financing
Rates of return on government securities depend, in part, on the size and growth of federal
government debt. If federal deficits are increasing over time, then the Treasury will be
constantly trying to raise funds in the financial market. As we discussed in Chapter 3, the
Treasury sells a number of short-term, intermediate, and long-term securities. Which
securities the Treasury uses to finance federal deficit affects not only the yield curve for
Treasury securities, but also the yield curve for corporate bonds, which are a substitute to
Treasury securities. For example, if the Treasury were to finance a deficit by selling shortterm Treasury securities, the price of the short-term government securities (T-bills) would
be pushed down, providing higher yields. In the corporate bond market, the higher rates on
short-term government securities would lead to a decrease in the demand for short-term
corporate securities, which, in turn, would lead to an excess supply in that market as shortterm corporate bondholders try to sell their corporate bonds in order to buy the higher
yielding Treasury securities. However, as bondholders try to sell their short-term corporate
bonds, the prices on such bonds would decrease, causing the rates on short-term corporate
bonds to rise. Thus, an increase in short-term government rates causes an upward shift in
the short-term bond demand curve (see Exhibit 6.2-3). At the initial YTM, this shift would
lead to an excess supply of short-term corporate bonds, which would cause short-term rates
to increase until a new equilibrium rate, rST*, is attained. Since the long-term market is
assumed to be independent of short-term rates, the total adjustment to the higher short-term
government rates would occur through the increase in short-term corporate rates. Moreover,
given a yield curve that is initially flat, as shown in Exhibit 6.2-3, Treasury financing with
short-term securities causes the corporate yield curve to become negatively-sloped. If the
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yield curve for Treasury securities is initially flat as well, then financing with short-term
securities would also cause the yield curves for Treasury securities to become negatively
sloped.1
In the above case, the Treasury financed the government deficit with short-term
securities. If they had financed the deficit with long-term securities, the impact would have
been felt in the long-term bond market. In this case, the corporate bond yield curve, as well
as the Treasury's yield curve, would tend to exhibit a positive slope. Other possible
scenarios can be analyzed: A government surplus, such as the one that occurred in the late
1990s, which the Treasury uses to buy up existing Treasury securities in order to reduce the
federal government’s outstanding debt; a decision by the Treasury to refinance more of its
existing debt with more short-term securities or to refinance with longer-term bonds; or a
deficit financed with both short-term and long-term bonds in equal proportions. Most of
these cases can be analyzed in terms of basic supply and demand models.
Case 3: Open Market Operations
The Board of Governors of the Federal Reserve System is responsible for establishing and
implementing monetary policy in the U.S.
One important monetary tool the Federal
Reserve (Fed) uses is an open market operation. To lower interest rates in order to stimulate
the economy, the Fed uses an expansionary open market operation (OMO).
In an
expansionary OMO, the Fed buys existing Treasury securities. As it buys the securities, the
1
If the corporate yield had been initially positively sloped, then the increase in short-term
Treasury rates would have caused the yield curve to become flatter. If the yield curve had
been initially negatively sloped, then the rate change would have caused the curve to
become even more negatively sloped. In general, we can describe such an impact as `having
a tendency to cause the yield curve to become negatively sloped.'
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demand and price of the securities rise and yields fall. In contrast, when the Fed is fighting
inflation, it may try to slow the economy down by increasing interest rates through a
contractionary OMO. Here the Fed sells some of its Treasury securities, pushing Treasury
security prices down and yields up.
A yield curve for Treasury securities is obviously influenced by open market
operations. For example, if the Fed is engaged in an expansionary OMO in which it buys
short-term T-bills, there would be a tendency for the yield curve to become positivelysloped as the Fed tries to buy short-term securities. On the other hand, in a contractionary
OMO in which the Fed sells short-term bills, there is a tendency for the Treasury yield curve
to become negatively-sloped.
In addition to affecting Treasury yield curves, open market operations also change the
yield curve for corporate securities through a substitution effect.
For example, an
expansionary OMO in which the Fed purchases short-term Treasury securities would tend to
cause the yield curve for corporate securities to become positively-sloped. That is, as the
rate on short-term Treasury securities decreases as a result of the OMO, the demand for
short-term corporate bonds would increase, causing higher prices and lower yields on the
short-term corporate securities. If corporate yield curves were initially flat, as shown in
Exhibit 11.2-4, then the expansionary OMO would result in a new positively-sloped yield
curve.
The opposite results would occur if the Fed implemented a contractionary OMO in
which it sold short-term Treasury securities. The Fed could also engage in expansionary
and contractionary OMOs by buying and selling long-term Treasury securities.
analysis of these cases is left to the reader.
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The
6.2.3 Weaknesses of MST
MST provides an economic foundation for explaining the shapes of yield curves; that is,
using supply and demand models, the impact of a number of economic activities on the term
structure of interest rates can be analyzed. The problem with MST is not in what it explains,
but what it fails to include. First, MST does not recognize that it is possible that the rate of
return on a bond in a particular maturity segment could increase to a level sufficient to
induce investors to move out of their preferred segment and buy the bond with the higher
rate in exchange for greater risk exposure. Secondly, MST does not take into account the
greater volatility inherent in long-term bonds. Finally, the theory fails to include the role of
expectations in determining the structure of interest rates. An investor with a two-year
horizon period, for example, might prefer a series of one-year bonds to a two-year bond, if
she expects relatively high yields on one-year bonds next year. If there are enough investors
with such expectations, they could have an effect on the current demands for one- and twoyear bonds. These three limitations of MST are addressed in the other theories of term
structures: the Preferred Habitat Theory, Unbiased Expectations Theory, and Liquidity
Preference Theory.
6.3 PREFERRED HABITAT THEORY
MST assumes that investors and borrowers have preferred maturity segments or habitats
determined by the maturities of their securities that they want to maintain. The Preferred
Habitat Theory (PHT) posits that investors and borrowers may stray away from desired
maturity segments if there are relatively better rates to compensate them. Furthermore, PHT
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asserts that investors and borrowers will be induced to forego their perfect hedges and shift
out of their preferred maturity segments when supply and demand conditions in different
maturity markets do not match.
To illustrate PHT, consider an economic world in which, on the demand side, investors
in corporate securities, on average, prefer short-term to long-term instruments, while on the
supply side, corporations have a greater need to finance long-term assets than short-term,
and therefore prefer to issue more long-term bonds than short-term. Combined, these
relative preferences would cause an excess demand for short-term bonds and an excess
supply for long-term claims and an equilibrium adjustment would have to occur. As
summarized in Exhibit 6.3-1, the excess supply in the long-term market would force issuers
to lower their bond prices, thus increasing bond yields and inducing some investors to
change their short-term investment demands. In the short-term market, the excess demand
would cause bond prices to increase and rates to fall, inducing some corporations to finance
their long-term assets by selling short-term claims. Ultimately, an equilibrium in both
markets would be reached with long-term rates higher than short-term rates, a premium
necessary to compensate investors and borrowers/issuers for the risk they've assumed.
As an explanation of term structure, the PHT would suggest that yield curves are
positively sloped if investors, on the average, prefer short-term to long-term investments and
borrowers/issuers prefer long to short. A priori, such preferences may be the case. That is,
investors may prefer short-term investments given that longer maturity bonds tend to be
more sensitive to interest rate changes or because there are more investors in the upper
middle-age class (with shorter investment horizons) than in the young adult or middle-age
class (with longer horizon periods). Borrowers also may have greater long-term than short-
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term financing needs and thus prefer to borrow long-term. Hence, one could argue that the
yield curve is positively sloped because investors' and borrowers' preferences make the
economy poorly hedged. Of course, the opposite case in which investors want to invest
more in long-term securities than short-term and issuers desire more short-term to long-term
debt is possible. Under these conditions the yield curve would tend to be negatively-sloped.
6.4 EXPECTATIONS THEORY
6.4.1 Unbiased Expectations Theory
Expectations theories try to explain the impact of investor expectations on the term structure
of interest rates. A popular model is the Unbiased Expectations Theory (UET). Developed
by Fredrick Lutz, UET is based on the premise that the interest rates on bonds of different
maturities can be determined in equilibrium where implied forward rates are equal to
expected spot rates.
To illustrate UET, consider a market consisting of only two bonds: a riskless one-year
pure discount bond and a riskless two-year pure discount bond, both with principals of
$1,000. Suppose that supply and demand conditions are such that both the one-year and
two-year bonds are trading at an 8% YTM. Also suppose that the market expects the yield
curve to shift up to 10% next year, but, as yet, has not factored that expectation into its
current investment decisions (see Figure 6.4-1). Finally, assume the market is risk-neutral,
such that investors do not require a risk premium for investing in risky securities (i.e., they
will accept an expected rate on a risky investment that is equal to the risk-free rate). To see
the impact of market expectations on the current structure of rates, consider the case of
investors with horizon dates of two years. These investors can buy the two-year bond with
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an annual rate of 8%, or they can buy the one-year bond yielding 8%, then reinvest the
principal and interest one year later in another one-year bond expected to yield 10%. In a
risk-neutral market, such investors would prefer the latter investment since it yields a higher
expected average annual rate for the two years of 9%:
E(YTM)  [(108
. )(110
. )]1/ 2  1  .09.
Similarly, investors with one-year horizon dates would also find it advantageous to buy a
one-year bond yielding 8% than a two-year bond (priced at $857.34 = $1,000/1.082), which
they would sell one year later to earn an expected rate of only 6%. That is:
$1000
 $857.34
(108
. )2
$1000
E( P1b,1 ) 
 $909.09
(110
. )1
$909.09  $857.34
E(1  yr rate) 
 .06.
$857.34
b
PMt
 P2b,0 
Thus, in a risk-neutral market with an expectation of higher rates next year, both investors
with one-year horizon dates and investors with two-year horizon dates would purchase
one-year instead of two-year bonds.
If enough investors do this, an increase in the demand for one-year bonds and a
decrease in the demand for two-year bonds would occur until the average annual rate on the
two-year bond is equal to the equivalent annual rate from the series of one-year investments
(or the one-year bond's rate is equal to the rate expected on the two-year bond held one
year). In the example, if the price on a two-year bond fell such that it traded at a YTM of
9%, and the rate on a one-year bond stayed at 8%, then risk-neutral investors with two-year
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horizon dates would be indifferent between a two-year bond yielding a certain 9% and a
series of one-year bonds yielding 10% and 8%, for an expected rate of 9%. Investors with
one-year horizon dates would likewise be indifferent between one-year bonds yielding 8%
and a two-year bond purchased at 9% and sold one year later at 10%, for an expected oneyear rate of 8%. Thus in this case, the impact of the market's expectation of higher rates
would be to push the longer-term rates up.
Recall that in the last chapter we defined the implied forward rate as a future rate
implied by today's rates. In this example, the equilibrium YTM on the two-year bond is 9%
and the equilibrium YTM on the one-year bond is 8%, yielding an implied forward rate of
10%, the same as the expected rate on a one-year bond, one year from now. That is:
YTM 2  (1  YTM1 )(1  f11 )
f11 
(1  YTM 2 ) 2
1
(1  YTM1 )
f11 
(109
. )2
 1  .10.
(108
. )
1/ 2
 1
Thus in equilibrium, the implied forward rate is equal to the expected spot rate.
6.4.2 Yield Curves that Incorporate Investors’ Expectations
In the above example, the yield curve is positively sloped, reflecting expectations of higher
rates. By contrast, if the yield curve were currently flat at 10% and there was a market
expectation that it would shift down to 8% next year, then the expectation of lower rates
would cause the yield curve to become negatively sloped (see Figure 6.4-2). In this case, an
investor with a two-year horizon date would prefer the two-year bond at 10% to a series of
one-year bonds yielding an expected rate of only 9% (E(YTM) = [(1.10)(1.08)]1/2 -1 = .09);
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a risk-neutral investor with a one-year horizon would also prefer buying a two-year bond
which has an expected rate of return of 12% to the one-year bond which yields only 10%.
In a risk-neutral market, the expectations of lower rates would cause the demand and price
of the two-year bond to increase, lowering its rate, and the demand and price for the oneyear bond to decrease, increasing its rate. These adjustments would continue until the rate
on the two-year bond equaled the average rate from the series of one-year investments, or
until the rate on the one-year bond equaled the expected rate from holding a two-year bond
one year (or when the implied forward rate is equal to expected spot rates). In this case, if
one-year rates stayed at 8%, then the demand for the two-year bond would increase until it
was priced to yield 9% - the expected rate from the series: [(1.10)(1.08)]1/2 -1 = .09 (see
Figure 6.4-2).
Finally, suppose the yield curve is currently positively-sloped, with the rate on the oneyear bond at 8% and the rate on the two-year bond at 9%. This time, though, suppose that
the market expects no change in yields and that this expectation has not yet been factored in.
Investors with horizon dates of two years would prefer the two-year bond yielding 9% to a
series of one-year bonds yielding 8% now and 8% expected at the end of one year (for a
two-year equivalent of 8%). Similarly, risk-neutral investors with one-year horizon dates
would also prefer the two-year bond held for one year in which the expected rate is 10%,
compared to only 8% from the one-year bond. Given the preference for two-year bonds, a
market response would occur in which demand for the two-year bond would rise, in turn
increasing its price and lowering its yield, and the demand and price for the one-year bond
would decrease, increasing its yield. Thus, the original positively-sloped yield curve would
become flat, given the expectation of no change in the yield curve.
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6.4.3 Features of UET
One of the features of UET is that in equilibrium the yield curve reflects current
expectations about future rates. From our preceding examples, when the equilibrium yield
curve was positively-sloped, the market expected higher rates in the future; when the curve
was negatively-sloped, the market expected lower rates; when it was flat, the market
expected no change in rates.
Secondly, UET intuitively captures what should be considered as normal market
behavior. That is, whether or not the market is risk neutral, has perfect expectations, or
bonds are perfect substitutes, investors, as well as borrowers/issuers do factor in
expectations. For example, if long-term rates were expected to be higher in the future
(based perhaps on the expectation of greater economic growth), long-term investors (e.g.,
life insurance company, pension fund, etc.) would not want to purchase long-term bonds
now, given that next period they would be expecting higher yields and lower prices on such
bonds (they also would be exposed to possible capital losses if they did buy such bonds and
were forced to liquidate them next year). Instead, such investors would invest in short-term
securities now, reinvesting later at the expected higher long-term rates.
In contrast,
borrowers/issuers wishing to borrow long-term would want to sell long-term bonds now
instead of later at possibly higher rates. Combined, the decrease in demand for long-term
bonds by investors and the increase in the supply of long-term bonds by borrowers would
serve to lower long-term bond prices and increase yields, leading to a positively-sloped yield
curve. Thus, a salient feature of UET is that it incorporates expectations as an important
variable in explaining the structure of interest rates.
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Finally, if UET strictly holds (i.e., we can accept all of the model's assumptions), then
the expected future rates would be equal to the implied forward rates. As a result, one could
forecast futures rates and future yield curves by simply calculating implied forward rates
from current rates (this technique is explained in Section 6.7).
6.5 LIQUIDITY PREFERENCE THEORY
The Liquidity Preference Theory (LPT), also referred to as the Risk Premium Theory
(RPT), posits that there is a liquidity premium for long-term bonds over short-term bonds.
Recall that in Chapter 5 we examined how long-term bonds were more price sensitive to
interest rate changes than short-term bonds. As a result, the prices of long-term securities
tend to be more volatile and therefore more risky than short-term securities. According to
LPT, if investors are risk averse, then they would require some additional return (liquidity
premium) in order to hold long-term bonds instead of short-term ones. Thus, if the yield
curve were initially flat, but had no risk premium factored in to compensate investors for the
additional volatility they assumed from buying long-term bonds, then the demand for longterm bonds would decrease and their rates increase until risk-averse investors were
compensated. In this case, the yield curve would become positively-sloped
6.6 SUMMARY OF THEORIES
As we noted earlier, the structure of interest rates cannot be explained in terms of any one
theory; rather, it is best explained by a combination of theories. Of the four theories, the two
major ones are MST and UET.
MST is important because it establishes how the
fundamental market forces governing the supply and demand for assets determines interest
19
rates. UET, in turn, extends MST to show how expectations impact the structure of interest
rates. PHT, by explaining how markets will adjust if the economy is poorly hedged, and
LPT, by including a liquidity premium for longer-term bonds, both represent necessary
extensions to MST and UET. Together, the four theories help us to understand how supply
and demand, economic conditions, government deficits, monetary policy, hedging, maturity
preferences, and expectations affect the bond market in general and the structure of rates in
particular.
6.7 EMPIRICAL YIELD CURVES: SPOT YIELD CURVE
In the past, many analysts constructed yield curves from yield data on Treasury securities or
different corporate bonds. The problem with generating yield curves in this way, though, is
finding a sample of bonds that are identical in all respects except maturity (i.e., the same
coupon rates, risk, option features, and the like). Today, the convention is to generate a spot
yield curve showing the relation between the spot rates and maturity. As noted in Chapter 5,
a spot Treasury rate is the rate on a pure discount Treasury bond. Since such bonds lack
coupons, have no default risk, and, if generated from T-bills, have no risk of call, they are
ideal.2 Once a spot yield curve is generated from Treasury securities, the yield curve for
bonds with different coupons and quality can be estimated by including appropriate
premiums. The obvious problem with generating spot yield curves from Treasury securities
is that there are no zero coupon Treasury securities with maturities greater than one year. As
noted in the last chapter, though, theoretical spot yield curve can be determined using a
2
Note, in the last chapter we argued that the correct way to value bonds is to discount
their cash flows by spot rates. If the market does not value bonds this way, then arbitrage
opportunities would exist by buying the bond and stripping it, or buying stripped securities
and rebundling them.
20
bootstrapping technique. Appendix 6A presents an example of how to construct a spot yield
curve using the bootstrapping technique.3
6.7.1 Forecasting Future Spot Yield Curves
Recall from our earlier discussion of the term structure of interest rates that one of the
implications of UET is that the current yield curve is governed by the condition that the
implied forward rate is equal to the expected spot rate. Given a spot yield curve, one could
therefore use UET to estimate next period's spot yield curve by determining the implied
forward rates.
Exhibit 6.7-1 shows spot rates on bonds with maturities ranging from one year to five
years (Column 2). From these rates, expected spot rates are generated for bonds one year
from the present (Column 3) and two years from the present (Column 4). The expected spot
rates shown are equal to their corresponding implied forward rates. For example, the
expected rate on a one-year bond one year from now (E(smt) = E(s11)) is equal to the implied
forward rate of fmt = f11 = 11%. This rate is obtained by using the geometric mean with the
current two-year and one-year spot rates:
s2  (1  s1 )(1  f11 )
1/ 2
f11 
(1  s2 ) 2
1
1  s1
f11 
(1105
. )2
 1  .11.
110
.
3
 1.
There are techniques other than bootstrapping that are used to estimate spot rates. For a
discussion, see Fabozzi (1995). Also note that stripped Treasury securities could also be
used to determine spot rate. Studies, though, have shown that Treasury stripped securities
often sell at rates different than theoretically generated spot rates.
21
Similarly, the expected two-year spot rate one year from now is equal to the implied forward
rate on a two-year bond purchased one year from now of fmt = f21 = 11.5%. This rate is
obtained using three-year and one-year spot rates (see Exhibit 6.7-1). The other expected
spot rates for next year are found repeating this process.
A similar approach also can be used to forecast the yield curve two years from the
present. The expected one-year spot rate two years from now, E(s12), is equal to the implied
forward rate f12, which is equal to 12%. This rate is obtained from two-year and three-year
spot rates using the geometric mean. The expected two-year spot rate, two years from now,
is found by solving for f22. Using four-year and two-year spot rates, f22 is equal to 12.5%.
Finally, f32 of 13% is found using five-year and two-year bonds (see Exhibit 6.7-1). All of
these implied forward rates can be found using the following formula:
f Mt 
M
(1  sMt ) M  t
(1  st ) t
 1.
6.7. 2 Using Forward Rates as Expected Rates of Return
According to UET, if the market is risk-neutral, then the implied forward rate is equal to the
expected spot rate, and in equilibrium, the expected rate of return for holding any bond for
one year would be equal to the current spot rate on one-year bonds. Similarly, the rate
expected to be earned for two years from investing in any bond or combination of bonds
(e.g., such as a series of one-year bonds) would be equal to the rate on the two-year bond.
This condition can be illustrated using the spot yield curve and expected yield curves (or
forward rates) shown in Exhibit 6.7-1. For example, the expected rate of return from
purchasing a two-year pure discount bond at the spot rate of 10.5% and selling it one year
22
later at an expected one-year spot rate equal to the implied forward rate of f11 = 11% is 10%.
This is the same rate obtained from investing in a one-year bond. That is:
E( r ) 
90.09  818984
.
 .10,
818984
.
where:
Face Value  100
100
 90.09
111
.
100

 818984
.
.
(1105
. )2
E( P11 ) 
P20
Similarly, the expected rate of return from holding a three-year bond for one year, then
selling it at the implied forward rate of f21 is also 10%. That is:
E( r ) 
80.43596  731191
.
 .10,
731191
.
where:
Face Value  100
100
 80.43596
(1115
. )2
100

 731191
.
.
(111
. )3
E( P21 ) 
P30
Any of the bonds with spot rates shown in Exhibit 6.7-1 would have expected rates for
one year of 10%, if the implied forward rate is used as the estimated expected rate. Similar
results hold for a two-year investment period. That is, any bond held for two years and sold
at its forward rate would earn the two-year spot rate of 10.5%. For example, a four-year
bond purchased at the spot rate of 11.5% and expected to be sold two years later at f22 =
12.5%, would trade at an expected rate of 10.5% - the same as the current two-year spot:
23
E( r ) 
79.0123
 1  .105,
64.6994
where:
Face Value  100
100
 79.0123
(1125
. )2
100

 64.6994.
(1115
. )4
E( P22 ) 
P40
Similarly, an investment in a series of one-year bonds at spot rates of s1 = 10.0% and f11 =
11%, yields a two-year rate of 10.5%:
E( r )  (110
. )(111
. )
1/ 2
 1  .105.
Finally, the same conditions also hold for coupon bonds which are valued at spot rates
and are expected to be sold at expected spot rates equal to the implied forward rates. For
example, a four-year, 10% coupon bond with a face value of 100 would be worth 95.762 if
it is discounted by the spot rates shown in Exhibit 6.7-1:
V0b 
10
10
10
110



 95.762.
2
3
110
.
(1105
. )
(111
. )
(1115
. )4
If that bond is expected to be sold one year later at the forward rates, its expected price
would be 95.3484, yielding a one-year expected rate of 10% - the same as the one-year spot
rate:
24
10
10
110


 95.3484
2
111
.
(1115
. )
(112
. )3
(95.3484  95.762)  10
E( r ) 
 .10.
95.762
E ( P) 
If the bond is held two years and expected to be sold at forward rates, then its expected price
would be 95.842. If one assumes the first coupon is reinvested at f11 = 11%, then the
expected rate for two years would be 10.5% - the same as the two-year spot rate:
10
110

 95842
.
112
.
(1125
. )2
(95842
.
 10(111
. )  10
E( r ) 
 1  .105.
95.762
E ( P) 
That expected rates from holding any bond (or combination) for M years are equal to
the rate on a M-year pure discount bond is a direct result of the risk-neutrality assumption of
UET. In reality, bond markets are not risk-neutral. Not surprisingly, studies by Fama
(1976) and others have shown that forward rates are not good predictors of future interest
rates. Analysts often refer to forward rates as hedgable rates, and most do not consider
forward rates a market consensus on expected future rates. The most practical use of
forward rates or expected spot yield curves generated from forward rates is that they provide
cut-off rates useful in evaluating investment decisions. For example, an investor with a
one-year horizon date should only consider investing in the two-year bond in our above
example, if she expected one-year rates one year later to be less than f11 = 11%; that is,
assuming she is risk-averse and wants an expected rate greater than 10%. Thus, forward
rates serve as a good cut-off rate for evaluating investments.
25
6.8 CONCLUSION
The key feature that distinguishes debt securities from equity securities is that debt
instruments have maturities. The term structure of interest rates describes the relationship
between the yields on debt securities and their maturity. As we have seen in this chapter, the
term structure can vary from a direct to an inverse relation. The four prominent theories
explaining term structure show that such factors as market expectations, economic
conditions, and risk-return preferences are all important in determining the structure of rates.
26
Exhibit 6.1-1
Types of Yield Curves
• Shapes:
YTM
Normal
Flat
Inverse
M
27
Exhibit 6.2-1
Supply and Demand Model for S-T and L-T
Bonds
• MST Model:
rST
rLT
S
BST
B D B D ( rSTG )
BST
r
D
S
BST
 BST
*
rLT
r
B LT
D
S
B LT
 B LT

*
ST
B D B D ( rLTG )

*
rLT

rST*
S
BLT
ST

Yield Curve
LT
M
Exhibit 6.2-2
Impact of an Economic Expansion on Corporate
Yield Curve
• MST Model:
rST
B
S
ST
rLT
S
B ST
'
D
D
*
) rLT '
*
rLT
G
ST
B B (r
*
ST
*
ST
r '
r

S
B LT


S
B LT
'
B D B D ( rLTG )

BST
Expansion  upward Shift
of YC
*
rLT
'
rST* '
B LT
r
*
rLT
rST*
28


ST


LT
Yield Curve
M
Exhibit 6.2-3
Impact of Treasury Sale of
ST Bonds on Corporate YC
• MST Model:
rST
S
BST


rST* '
*
ST
r
rLT
B D B D ( rSTG  )
B D B D ( rSTG )
S
B LT
*
rLT
B D B D ( rLTG )

BST
B LT
r
Treasury Sale of ST Bond
 YC rotates up


ST
rST* '
*
rST*  rLT
Yield Curve

LT
M
Exhibit 6.2-4
Expansionary OMO with ST Bonds
• MST Model:
rST
S
BST
rST*
rST* '


B D B D ( rSTG  )
r
*
*
Fed buys ST government bonds rST  rLT
 YC rotates down
rLT
B D B D ( rSTG )
rST* '
S
B LT

*
rLT
BST
B LT



ST
29
B D B D ( rLTG )
LT
Yield Curve
M
Figure 6.4-2
UET with Expectation of Higher Rates
• Investors with HD of 2 years and those with HD
of 1 year would prefer one-year bonds over twoyear bonds.
• Market Response:
B2D   P2B   r2 
r
10%
8%


1 yr
YC becomes
B1D   P1B   r1 


UET
Positively Sloped
M
2 yr
• Assume that the market response is one in
which only the demand for 2-year bonds is
affected by the expectations.
BD2 P2B r2 
r2 until r2 r2:Series.09
r
10%
9%
8%


1 yr
Note: When YTM 2  9%, YTM 1  8% ,
then f 11  10%  E ( r11 ).
Thus, if UET holds, then f Mt  E ( rMt )


UET
M
2 yr
30
Exhibit 6.3-1
Preferred Habitat Theory
• Poorly Hedged Economy: Investors, on average,
prefer ST investments; corporate borrowers, on
average, prefer to borrow LT (sell LT corporate
bonds:
Investors
prefer ST
Borrower
prefer LT
Excess Demand in ST

Excess Supply in LT
31
 PSBT  , rS T  
rST  attracts
LT Borrowers

P LBT  , r L T 

rLT  attracts
ST Investors
Figure 6.4-3
UET with Expectation of Lower Rates
• Investors with HD of 2 years and those with HD
of 1 year would prefer two-year bonds over oneyear bonds.
• Market Response:
B D2   P2B   r2 
r
10%
8%


1 yr
B1D  P1B  r1 
YC becomes
Negatively Sloped


UET
M
2 yr
• Assume that the market response is one in
which only the demand for 2-year bonds is
affected by the expectations.
BD2 P2B r2 
Note :When YTM 2  9%, YTM 1  10 %,
then f11 8%  E ( r11 ).
r2  until r2  r2:Series.09
r
10%
9%
8%


1 yr
Thus , if UET holds , then f11 E ( r11 )


UET
M
2 yr
32
Exhibit 6.7-2
Forecasting Yield Curves Using Implied Forward Rates
(1)
(2)
(3)
(4)
Maturity
Spot Rates
Expected Spot Rates
One Year From
Present
Expected Spot Rates
Two Years from
Present
1
2
3
4
5
10.0%
10.5
11.0
11.5
12.0
f11 = 11%
f21 = 11.5%
f31 = 12.0%
f41 = 12.5%
f12 = 12%
f22 = 12.5%
f32 = 13%
f12
s3  (1  s1 )(1  f11 )(1  f12 )
1/ 3
s3  (1  s2 ) (1  f12 )
2
f12
1/ 3
f22
1
s4  (1  s1 )(1  f11 )(1  f12 )(1  f13 )
1
s4  (1  s2 ) 2 (1  f22 ) 2
(1  s3 ) 3

1
(1  s2 ) 2
f12 
(111
. )3
 1  .12.
(1105
. )2
f22 
(1  s4 ) 4
1
(1  s2 ) 2
f22 
(1115
. )4
 1  .125.
(1105
. )2
f32
s5  (1  s1 )(1  f11 )(1  f12 )(1  f13 )(1  f14
s5  (1  s2 ) 2 (1  f32 ) 3
f32 
3
f32  3
1/ 5
(1  s5 ) 5
1
(1  s2 ) 2
(112
. )5
 1  .13.
(1105
. )2
33
1/ 4
1
1/ 5
1
1
1/ 4
1
APPENDIX 6A
CONSTRUCTING A THEORETICAL SPOT YIELD CURVE USING
BOOTSTRAPPING
The bootstrapping technique described in Chapter 5 requires taking at least one pure
discount bond and then sequentially generating other spot rates from coupon bonds. To
illustrate, consider the Treasury securities presented in Table 6.A-1. As shown, there are
two T-bills with maturities of six months (.5 years) and one year, trading at annualized
yields of 5% and 5.25%. Since T-bills are pure discount bonds, these rates can be used as
spot rates (st) for maturities of .5 years (s.5) and one year (s1). To obtain the spot rates for 1.5
years (18 months) we first take the T-note with a maturity of 1.5 years, annual coupon rate
of 5.5% (semi-annual coupons of 2.75), and currently priced at par; next we value that bond
by discounting its cash flows at spot rates; finally, we solve for the spot rate for 1.5 years.
Doing this yields a spot rate of s1.5 = 5.51%. That is:
P1.5 
CF.5
b1  (s
.5
100 
/ 2)
CF1.0


CF1.5
g b1  (s / 2)g b1  (s
1
2
1
1.5
/ 2)
g
3
2.75
2.75
102.75


1
2
(1  (.05 / 2)
(1  (.0525 / 2)
(1  (s1.5 / 2) 3
94.705956 
102.75
b1 (s / 2)g
LL 102.75 O 1O
M
2  .0551.
P
M
P
94
.
705956
N
Q
M
P
N
Q
3
1.5
1/ 3
s1.5
To obtain the spot rate for a two-year bond (s2) we repeat the process using the two-year
bond paying semi-annual coupons of 2.875 and selling at par. This yields a spot rate of s2 =
5.77%:
34
P2 
CF.5
b1  (s
.5
100 
/ 2)
CF1.0


CF1.5
g b1  (s / 2)g b1  (s
1
2
1
1.5
/ 2)

CF2
g b1  (s / 2)g
3
4
2
2.875
2.875
2.875
102.875



1
2
3
(1  (.05 / 2)
(1  (.0525 / 2)
(1  (.0551 / 2)
(1  (s2 / 2) 4
91815421
.

102.875
b1 (s / 2)g
LL102.875 O 1O
M
2  .0577.
M
P
.
N91815421
Q P
M
P
N
Q
4
2
1/ 4
s2
Continuing the process with the other securities in Table 6.7-1, we obtain spot rates for
bonds with maturities of 18 months and two years of s2.5 = 6.031% and s3 = 6.290%.
35
Table 6.A-1
Treasury Securities Used For
Estimating Spot Rates
Security
Type
Maturity
Semi-Annual
Coupon
Annualized
YTM
Face
Value
Current
Price
1
T-Bill
.5 years
-
5%
100
90
2
T-Bill
1 year
-
5.25%
100
94.9497
3
T-Note
1.5 years
2.75
5.5%
100
100
4
T-Note
2 years
2.875
5.75%
100
100
5
T-Note
2.5 years
3
6.00%
100
100
6
T-Note
3 years
3.1252
6.25%
100
100
SPOT YIELD CURVE
MATURITY
SPOT RATES
.5 years
1 year
1.5 years
2 years
2.5 years
3 years
5%
5.25%
5.551%
5.577%
6.031%
6.290%
st
6.25
6.00
5.75
5.5
5.25
5
.5
1.0
1.5
2
36
2.5
3
M